Let us recall a fragment from an interview given by Harvey Friedman to a
"journalist" (full text at
http://www.cs.nyu.edu/pipermail/fom/2003-October/007509.html):
...
>> ... The imaginary natural number system
>> enjoys some very fundamental properties that the imaginary set system
does
>> not.
>>>> It seems so. I would wish a more detailed explanation of this for
students.
> Are these fundamental properties somehow impacted by the sad fact that
every
> such explanation involves "rules", i.e. a structure possibly equivalent to
> natural numbers? Or, are these properties completely implied by this fact?
> How to get over this feeling?
For students:
The natural numbers under successor has no proper substructure.
Note that
The set theoretic universe under membership has a proper substructure.
E.g., remove the empty set.
However, one is naturally lead to considering statements like this.
1. The set theoretic universe has a proper elementary substructure.
...
End of quote.
How to explain this "privileged" position of natural numbers among other
mathematical structures? Couldn't it be caused by the "sad fact" that any
formalization of any mathematical structure involves some formal syntax,
i.e. - something very similar to a (or, "the"?) natural number system?
If we are trying to formalize structures by means of formal syntax, then the
structures that are equivalent to this formal syntax, should look "better"
than other structures?
Doesn't this phenomenon re-occur in the model theory of non-countable
languages?
Best wishes,
Karlis.Podnieks at mii.lu.lv
Institute of Mathematics and Computer Science
University of Latvia